Introduction to Statistical Methods
Department of Educational Psychology
We often aim to calculate the Area-under-the-curve (AUC) using the cumulative distribution function (cdf).
The cumulative distribution function follows these rules:
Compared to the tables of discrete probabilities, it is usually easier to conceptualize continuous probabilities using a histogram
For more basic examples: basic geometry of width * height works
The following examples are simpler to understand and visualize because the “curve” of \(f(x)\) here is just a straight line.
It is often useful to start with simply identifying the range of values, from the minimum possible value to the maximum possible value
\(f(x) = \frac{1}{20}\) for event of \(0 \leq x \leq 20\)
Using normal geometry: \(AREA = (20 - 0)(\frac{1}{20}) = 1.00 \rightarrow\) 100% probability of \(X\) falling between 0 and 20
\(f(x) = \frac{1}{20}\) for event of \(0 \leq x \leq 2\)
Using geometry again: \(AREA = (2 - 0)(\frac{1}{20}) = 0.10 \rightarrow\) 10% probability of \(X\) falling between 0 and 2
\(f(x) = \frac{1}{20}\) for event of \(4 \leq x \leq 15\)
Using geometry again: \(AREA = (15 - 4)(\frac{1}{20}) = 0.55 \rightarrow\) 55% probability of \(X\) falling between 0 and 2
Like with discrete data, we can estimate probabilities of certain data occurring, but take a more visual approach
We will cover three distinct, different patterns of continuous probability functions, just like we did with the binomial distribution in the last unit
The uniform distribution starts with the assumption that all points within a range are equally likely to occur
So far, all of the examples prior have shown a uniform distribution, as indicated by the straight horizontal line, parallel to the x-axis
It follows the notation: \(X \sim U(a,b)\) where:
From here, we can construct a probability density function (pdf) as \(f(x) = \frac{1}{b - a}\)
Like with discrete random variables, we can do estimates for the expected long-term mean and standard deviation
Mean: \(\mu = \frac{a + b}{2}\)
Standard Deviation: \(\sigma = \sqrt{\frac{(b - a)^2}{12}}\)
Mean:
\[ \mu = \frac{90 + 120}{2} = 105 \]
Standard Deviation:
\[ \sigma = \sqrt{\frac{(120 - 90)^2}{12}} = 8.66 \]
Continuous variables are common outcomes and predictors in educational and social science research - it is important we can accurately describe the probability and structure of this data
Much like with discrete random variables, we can represent continuous random variables with probability functions, albeit with a different procedure than the tabular format of probability distribution functions (PDF) as introduced in module 4
Just like we learned about the binomial distribution pattern for discrete variables, we can apply specific patterns for continuous random variable as well: here, we covered the uniform and exponential distributions which can be helpful in understand some natural continuous variables. These patterns are “ideal” tools that we can use in analyzing our data.
However, we still have yet to cover arguably the most important pattern of continuous distribution, the normal distribution, which will be covered more in the next chapter.
Module 5 Lecture - Continuous Random Variables || Introduction to Statistical Methods